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\title{基于球形支持向量机的多视角学习研究}
\author{作者}
%\date{\today}
% 汉语标题
\bjfuTitle{请在此写上你的论文标题}
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\bjfuTitleEn{Please text your english title here}


% 汉语摘要
\bjfuAbstract{
请在此处撰写摘要
}

% 英语摘要
\bjfuAbstractEn{

Please text your english abstract.
}

% 汉语关键字
\bjfuKeywords{ 支，持，向，量，机}

% 英语关键字
\bjfuKeywordsEn{Sup, port vec,tor, mach, ine}
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\begin{document}
\makeBjfuTitlePage
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\section{章标题}\label{sec: introduction}
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\subsection{节标题}\label{sec: background}
支持向量机(support vector machine, SVM)\citep{svm_Vapnik1995}是机器学习领域最经典
的算法之一，它由Vapnik等人在1995年正式发表。SVM通过寻找一条以最大间隔分开训练样本的超
平面来实现对数据的判别。引入核方法\citep{kernel_method_1999advances}后，SVM可通过在
特征空间中构建超平面实现对原空间中数据的非线性判别。SVM的学习目标兼顾了间隔最大化和结
构风险最小化\citep{margin_1992}，这使得SVM拥有对未知数据良好的泛化性能和可靠的理论支
撑。SVM在小数据集、非线性和高维数据分类问题上表现优异，且在诸多交叉学科领域取得了成功
的应用，比如模式识别
\citep{lopez2016svmpattern5,wu2018svmpattern2}、文本分类
\citep{peng2019svmtext3}和生物信挖掘
\citep{liu2020svmbio,zhang2020svmbio3}等等。
...\\


\noindent \textbf{协同训练} \quad 
你可以这样。\\

\noindent \textbf{协同正则化} \quad 实现一个。\\

\noindent \textbf{间隔一致化} \quad 简单的罗列。\\


本研究的主要创新点和贡献可总结为以下五点：
\begin{itemize}%[$\bullet$]
	\item[$\bullet$] 也可以通过；
	\item[$\bullet$] 
	itemize实现；
	\item[$\bullet$] 罗列；
\end{itemize}

\section{第二章标题}
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\subsection{表格插入的参考样式}
为了简便起见，\autoref{tab: notations}总结了本论文出现的主要数学符号。

%\linespread{1.4}
\begin{table*}[h]
	\renewcommand\arraystretch{1.25}
	\centering
	\bicaption{符号对照说明表 \vspace{1.1ex}}{Table of notations}
	\label{tab: notations}
	\begin{tabular}{p{160pt}p{260pt}}
		\toprule[1.2pt]
		\textbf{符号} & \textbf{说明} \\
		\midrule
		$(x_{i}, y_{i})$ & 第$i$个单视角训练样本及其标签 $y_{i}$\\
		$(x_{i}^{A}, x_{i}^{B}, y_{i})$ & 第$i$个多视角训练样本及其标签 $y_{i}$\\
		$l^{+}$, $l^{-}$ & 正类(负类)训练样本的个数 \\
		$\boldsymbol{c}_{\pm}^{A},  \  \boldsymbol{c}_{\pm}^{B}$ & 
		分别对应于视角A和视角B的正类(负类)超球球心 \\
		$R_{\pm}^{A} , \ R_{\pm}^{B} $ & 
		分别对应于视角A和视角B的正类(负类)超球半径 \\
		$ \langle x_{i}, \ x_{j} \rangle$ & 
		向量$x_{i}$和$x_{j}$的内积，或写为$x_{i}^{\top}x_{j}$ \\
		$\varphi( \ \cdot \ ), \  \varphi_{A}( \ \cdot \ ), \   \varphi_{B}( \ 
		\cdot \ )$  & 将原空间
		映射到高维特征空间的正定核映射 \\
		$k(x_{i}, x_{j})$ & 传统单视角中的核函数 $ \langle 
		\varphi(x_{i}), \ \varphi(x_{j}) \rangle$ \\
		$k^{A}(x_{i}^{A}, x_{j}^{A})$ & A视角核函数 $ \langle 
		\varphi_{A}(x_{i}^{A}), \ \varphi_{A}(x_{j}^{A}) \rangle$\\
		$ k^{B}(x_{i}^{B}, x_{j}^{B})$ & 
		B视角核函数$ \langle
		\varphi_{B}(x_{i}^{B}), \ \varphi_{B}(x_{j}^{B}) \rangle $\\
		$\left\| \ \cdot \ \right\|_{p}$ & 向量的$l_p$-范数 
		(如果$p$省略，则默认取$l_2$-范数) \\
		$\boldsymbol{Q}_{(\cdot \times \cdot)}$ & 大写黑体西文字符表示矩阵，其规模
		可
		见下标 \\
		$\boldsymbol{q}_{(\cdot \times \cdot)}$ & 小写黑体西文字符表示列向量，其长
		度可见下标 \\
		$diag(\cdot)$ & 表示取矩阵对角线元素的运算符\\
		\bottomrule[1.2pt]
	\end{tabular}
\end{table*}
\subsection{公式输入的参考样式}\label{sec: svm}
支持向量机(SVM)是机器学习的经典算法，它的优化学习目标是在两类样本点之间寻求最大间隔和
最小误
差的权衡。令$\mathscr{X} \in 
{{R}^{{{d}}}}$为$d$维的样本属性空间，$\mathscr{Y}=\left\{ 
-1,+1 
\right\}$为样本标签空间。考虑一个有监督二分类问题，其训练样本集表示为$T=\left\{
\left( x_{i},{{y}_{i}} \right)\left| x_{i}\in \mathscr{X},{{y}_{i}}\in 
\mathscr{Y} \right. \right\}_{i=1}^{l}$，SVM将在样本空间中寻找能将两类样本点以最大
间隔分开的决策超平面：
\begin{equation}
	\langle \boldsymbol{\omega}, \varphi (x) \rangle + b = 0 ,
\end{equation}
其中，${\boldsymbol{\omega}}$和$b$分别表示样本属性空间中决策超平面的法向量与
偏置项。

SVM的原问题是：
\begin{equation}
	\begin{aligned}
		\underset{\boldsymbol{\omega},b,\xi}{\mathop{\min }} & \quad 
		\cfrac{1}{2}{{\left\| \boldsymbol{\omega} 
		\right\|}^{2}} + C \sum\limits_{i=1}^{l}{{{\xi }_{i}}} \\ 
		s.t. & \quad {{y}_{i}}\left( \langle \boldsymbol{\omega}, \varphi 
		(x_{i}) 
		\rangle + b \right)\le 1-{{\xi }_{i}}, \\
		& \quad \xi_{i} \ge 0, \ i = 1,2,...,l,
	\end{aligned}
\end{equation}
其中，${{\varphi}(.)}$表示任意正定核映射，$C$是松弛变量的惩罚参数，且$C>0$。

在SVM的目标函数中，正则化项$\cfrac{1}{2}{{\left\| \boldsymbol{\omega} 
\right\|}^{2}}$旨在最大化超平面$\langle \boldsymbol{\omega}, \varphi (x) \rangle 
+ b = +1$和$\langle \boldsymbol{\omega}, \varphi (x) \rangle + b = 
-1$之间的距离，起到结构风险最小化的作用；而约束条件要求样本点在两条超平面之外，若落入
超平面之间，则引入松弛变量$\left\{ {{\xi }_{i}} 
\right\}_{i=1}^{l}$来使得约束成立；目标函数的第二项是最小化松弛变量之和，旨
在尽可能地避免样本落入间隔内，起到经验风险最小化的作用；惩罚因子$C$用于权衡目标函数的
结构风险和经验风险。

SVM的对偶问题是：
\begin{equation}
	\begin{aligned}
		\underset{\alpha}{\min} & \quad 
		\cfrac{1}{2}\underset{i=1}{\overset{l}{\mathop \sum }} 
		\underset{j=1}{\overset{l}{\mathop \sum }}{{\alpha }_{i}}{{\alpha 
			}_{j}}{{y}_{i}}{{y}_{j}}k(x_{i},x_{j}) - 
		\underset{i=1}{\overset{l}{\mathop \sum }}{{\alpha }_{i}} \\
		s.t. & \quad \underset{i=1}{\overset{m}{\mathop \sum }}{{\alpha 
			}_{i}}{{y}_{i}}=0,  \\
		& \quad 0 \le {{\alpha }_{i}}\le C, \ i=1,2,\ldots ,l.  \\
	\end{aligned} 
\end{equation}
通过对偶问题求解出$\boldsymbol{\omega}$和$b$后，可得到SVM对未知样本$x$的决策函数：
\begin{equation}
	f(x) = sign\left(\langle \boldsymbol{\omega}, \varphi(x) \rangle + b\right).
\end{equation}

\subsection{图片插入的参考样式}
SVM有着优良的分类性能，从\autoref{fig: demo_of_SVM}中可以看出，SVM能够准确地识别出不
同类
别的数据，并给出合理的的决策边界(图中黑色实线)。
\begin{figure}[h]
	\center
	% Use the relevant command to insert your figure file.
	% For example, with the graphicx package use
	\subcaptionbox{\text{传统软间隔SVM(线性核)} \label{fig: demo_of_linear_SVM}}
	{
		\includegraphics[width = 0.45\textwidth]{demo_of_linear_SVM_cropped.pdf}
	}
	\subcaptionbox{\text{非线性软间隔SVM(高斯核)} \label{fig: 
	demo_of_kernel_SVM}}
	{
		\includegraphics[width = 
		0.458\textwidth]{demo_of_kernel_SVM_cropped.pdf}
	} 
	\bicaption{线性SVM和非线性SVM在不同数据类型下的决策边界 
	\vspace{1.1ex}}{Classification 
	boundaries of linear SVM and nonlinear SVM under different data 
	distributions}
	\label{fig: demo_of_SVM}       % Give a unique label
\end{figure}


\subsection{长表格插入的参考样式}
\newcommand{\tabincell}[2]{\begin{tabular}{@{}#1@{}}#2\end{tabular}} 
{\zihao{5}
\begin{flushleft}
	\begin{longtable}[h]{p{55pt}p{50pt}p{50pt}p{50pt}p{50pt}p{50pt}p{50pt}}
		\bicaption{AwA数据集性能对比(Accu. $\pm$ std.) \vspace{1.1ex}}{Benchmark 
		results on AwA 
		data sets (Accu. $\pm $ std.)}
		\label{tab: AwA_res}\\
		\toprule[1.0pt]
		\endfirsthead
		\toprule[1.0pt]
		\multicolumn{7}{l}{续 \autoref{tab: AwA_res}}\\
		\midrule
		\endhead
		\bottomrule[1.0pt]
		\endfoot
		\tabincell{l}{\textbf{数据集名称} \\ {}} & \tabincell{l}{\textbf{SVM-2K} 
		\\ Accu. (\%) } & 
		\tabincell{l}{\textbf{MVMED} \\ Accu. (\%) } &  
		\tabincell{l}{\textbf{MvTwSVMs} \\ Accu. 
		(\%) } &
		\tabincell{l}{\textbf{MvNPSVM} \\ Accu. (\%) }& 
		\tabincell{l}{\textbf{Ours} \\ 
		\textbf{Accu. (\%)} } &
		\tabincell{l}{\textbf{Ours-2C} \\ \textbf{Accu. (\%)} }\\
		\midrule
		chi vs. pan & 68.15$\pm$4.81 & 47.99$\pm$4.48 & 68.42$\pm$2.31 & 
		85.57$\pm$0.83 & 86.56$\pm$0.71 & \textbf{88.70$\pm$0.68} \\
		chi vs. leo & 75.13$\pm$4.40 & 47.40$\pm$2.91 & 76.40$\pm$0.99 & 
		84.39$\pm$1.64 & 85.09$\pm$0.77 & \textbf{87.03$\pm$0.62} \\
		chi vs. per & 65.43$\pm$2.60 & 83.74$\pm$0.72 & 75.70$\pm$1.03 & 
		\textbf{85.10$\pm$0.78} & 84.13$\pm$0.71 & 84.29$\pm$0.61 \\
		chi vs. pig & 78.33$\pm$2.47 & 47.18$\pm$3.31 & 66.94$\pm$2.02 & 
		82.34$\pm$0.56 & \textbf{83.81$\pm$1.04} & 83.03$\pm$0.74 \\
		chi vs. hip & 45.58$\pm$1.81 & 49.57$\pm$4.41 & 71.71$\pm$1.69 & 
		81.70$\pm$1.12 & 80.88$\pm$0.66 & \textbf{84.07$\pm$1.12} \\
		chi vs. hum & 90.73$\pm$2.14 & 94.81$\pm$0.56 & 88.10$\pm$0.53 & 
		94.88$\pm$0.54 & 93.82$\pm$0.65 & \textbf{95.94$\pm$1.03} \\
		chi vs. rac & 83.43$\pm$1.33 & 46.86$\pm$2.81 & 63.65$\pm$1.65 & 
		83.31$\pm$0.75 & 84.69$\pm$0.99 & \textbf{84.95$\pm$1.11} \\
		chi vs. rat & 64.90$\pm$1.59 & 76.18$\pm$1.01 & 66.41$\pm$1.95 & 
		79.48$\pm$1.08 & \textbf{81.90$\pm$1.10} & 79.73$\pm$0.96 \\
		chi vs. seal & 82.05$\pm$1.37 & 84.04$\pm$0.72 & 74.51$\pm$1.25 & 
		47.68$\pm$3.64 & 86.49$\pm$0.76 & \textbf{88.14$\pm$0.58} \\
		pan vs. leo & 82.00$\pm$2.36 & 82.46$\pm$0.66 & 69.17$\pm$1.83 & 
		\textbf{88.97$\pm$0.97} & 87.07$\pm$0.89 & 88.09$\pm$0.64 \\
		pan vs. per & 56.90$\pm$0.76 & 87.73$\pm$0.76 & 80.56$\pm$1.10 & 
		89.27$\pm$0.74 & \textbf{89.45$\pm$0.71} & 88.17$\pm$0.35 \\
		pan vs. pig & 63.21$\pm$1.29 & 77.91$\pm$1.66 & 61.43$\pm$2.13 & 
		82.53$\pm$0.99 & \textbf{82.65$\pm$0.74} & 80.79$\pm$1.24 \\
		pan vs. hip & 75.56$\pm$0.78 & 83.65$\pm$0.75 & 68.94$\pm$1.79 & 
		88.51$\pm$1.03 & 85.72$\pm$0.98 & \textbf{89.04$\pm$0.66} \\
		pan vs.hum & 49.22$\pm$4.23 & 96.03$\pm$0.49 & 89.00$\pm$1.14 & 
		96.26$\pm$0.33 & 96.14$\pm$0.34 & \textbf{97.07$\pm$0.47} \\
		pan vs. rac & 76.75$\pm$2.50 & 84.56$\pm$1.14 & 61.44$\pm$1.00 & 
		87.26$\pm$1.19 & \textbf{87.90$\pm$1.00} & 87.31$\pm$0.91 \\
		pan vs. rat & 64.26$\pm$1.06 & 82.62$\pm$1.22 & 66.83$\pm$2.66 & 
		85.40$\pm$0.99 & \textbf{88.00$\pm$0.93} & 86.21$\pm$1.05 \\
		pan vs. sea & 80.50$\pm$1.40 & 87.06$\pm$0.64 & 74.77$\pm$2.19 & 
		88.59$\pm$0.84 & 88.66$\pm$0.66 & \textbf{90.23$\pm$0.70} \\
		leo vs. per & 52.16$\pm$4.36 & 50.66$\pm$7.14 & 85.70$\pm$0.76 & 
		\textbf{90.91$\pm$0.69} & 90.56$\pm$0.31 & 90.07$\pm$0.45 \\
		leo vs. pig & 65.01$\pm$1.10 & 47.10$\pm$3.70 & 72.45$\pm$1.28 & 
		78.19$\pm$0.87 & \textbf{78.99$\pm$1.11} & 78.42$\pm$1.21 \\
		leo vs. hip & 73.44$\pm$0.86 & 82.42$\pm$0.87 & 78.38$\pm$1.03 & 
		82.94$\pm$1.12 & 84.14$\pm$1.06 & \textbf{84.37$\pm$0.58} \\
		leo vs. hum & 85.05$\pm$3.77 & 95.57$\pm$0.61 & 91.65$\pm$0.29 & 
		95.25$\pm$0.40 & 95.13$\pm$0.19 & \textbf{95.94$\pm$0.25} \\
		leo vs. rac & 56.29$\pm$5.19 & 48.31$\pm$3.56 & 60.82$\pm$3.52 & 
		\textbf{77.32$\pm$1.46} & 75.29$\pm$1.11 & 76.39$\pm$1.46 \\
		leo vs. rat & 59.85$\pm$0.84 & 83.91$\pm$0.64 & 76.71$\pm$1.40 & 
		75.15$\pm$6.60 & 83.13$\pm$0.56 & \textbf{83.96$\pm$0.89} \\
		leo vs. sea & 62.94$\pm$4.96 & 87.26$\pm$0.67 & 84.37$\pm$0.71 & 
		\textbf{88.46$\pm$0.52} & 88.24$\pm$0.84 & 87.72$\pm$1.14 \\
		per vs. pig & 66.00$\pm$6.37 & 78.24$\pm$0.84 & 73.93$\pm$0.96 & 
		79.06$\pm$0.99 & \textbf{79.50$\pm$1.03} & 79.24$\pm$1.63 \\
		per vs. hip & 72.66$\pm$3.78 & 83.85$\pm$0.79 & 78.37$\pm$1.00 & 
		85.66$\pm$0.50 & 86.02$\pm$1.21 & \textbf{86.54$\pm$1.15} \\
		per vs. hum & 91.89$\pm$0.88 & 88.46$\pm$0.86 & 83.42$\pm$0.78 & 
		92.19$\pm$0.42 & 91.41$\pm$0.29 & \textbf{95.91$\pm$0.42} \\
		per vs. rac & 47.42$\pm$5.11 & 47.78$\pm$3.58 & 73.39$\pm$1.40 & 
		85.84$\pm$0.69 & 84.06$\pm$0.75 & \textbf{87.43$\pm$0.83} \\
		per vs. rat & 66.52$\pm$3.26 & 68.18$\pm$1.34 & 61.35$\pm$1.70 & 
		68.27$\pm$1.69 & 69.29$\pm$0.78 & \textbf{69.76$\pm$1.28} \\
		per vs. sea & 78.52$\pm$1.82 & 84.72$\pm$1.08 & 75.11$\pm$0.82 & 
		\textbf{85.30$\pm$1.17} & 84.01$\pm$0.76 & 83.35$\pm$0.68 \\
		pig vs. hip & 71.36$\pm$2.02 & 70.94$\pm$1.44 & 64.53$\pm$1.19 & 
		71.87$\pm$0.82 & 72.29$\pm$1.14 & \textbf{72.43$\pm$1.29} \\
		pig vs. hum & 90.11$\pm$1.29 & 91.26$\pm$0.67 & \textbf{94.15$\pm$0.40} 
		& 93.01$\pm$0.42 & 91.50$\pm$0.40 & 92.76$\pm$0.51 \\
		pig vs. rac & 72.32$\pm$1.94 & 73.88$\pm$1.49 & 58.46$\pm$0.81 & 
		80.98$\pm$1.36 & \textbf{80.96$\pm$0.85} & 80.07$\pm$1.20 \\
		pig vs. rat & 57.73$\pm$2.40 & 48.59$\pm$4.62 & 55.90$\pm$2.55 & 
		68.89$\pm$1.96 & 71.46$\pm$1.39 & \textbf{72.22$\pm$0.85} \\
		pig vs. sea & 75.08$\pm$0.83 & 78.03$\pm$1.02 & 67.10$\pm$2.11 & 
		\textbf{78.97$\pm$0.91} & 77.71$\pm$1.14 & 78.70$\pm$0.72 \\
		hip vs. hum & 81.40$\pm$1.05 & 88.15$\pm$1.07 & 75.67$\pm$1.54 & 
		\textbf{88.45$\pm$0.87} & 88.04$\pm$0.47 & 88.36$\pm$0.74 \\
		hip vs. rac & 79.20$\pm$1.11 & 79.61$\pm$0.83 & 69.54$\pm$1.23 & 
		\textbf{81.60$\pm$1.12} & 81.57$\pm$1.03 & 80.53$\pm$1.31 \\
		hip vs. rat & 59.95$\pm$0.85 & 47.85$\pm$2.67 & 67.42$\pm$2.82 & 
		75.01$\pm$1.07 & \textbf{79.08$\pm$1.21} & 77.19$\pm$0.97 \\
		hip vs. sea & 54.37$\pm$4.15 & 70.24$\pm$1.09 & 60.20$\pm$2.99 & 
		\textbf{70.95$\pm$1.35} & 69.85$\pm$0.90 & 69.96$\pm$1.26 \\
		hum vs. rac & 63.09$\pm$4.41 & 90.90$\pm$0.66 & 86.24$\pm$1.17 & 
		92.34$\pm$0.50 & 92.69$\pm$0.56 & \textbf{93.81$\pm$0.57} \\
		hum vs. rat & 84.48$\pm$0.67 & 88.22$\pm$0.67 & 81.62$\pm$0.81 & 
		\textbf{89.94$\pm$0.55} & 87.81$\pm$0.76 & 89.27$\pm$0.56 \\
		hum vs. sea & 84.91$\pm$1.54 & 83.99$\pm$0.63 & 74.72$\pm$1.43 & 
		85.08$\pm$0.72 & 85.28$\pm$0.86 & \textbf{85.37$\pm$0.59} \\
		rac vs. rat & 57.80$\pm$1.62 & 71.74$\pm$0.91 & 65.97$\pm$1.47 & 
		46.78$\pm$2.09 & \textbf{76.12$\pm$1.13} & 76.10$\pm$1.10 \\
		rac vs. sea & 52.77$\pm$5.45 & 85.72$\pm$0.79 & 76.25$\pm$1.34 & 
		\textbf{86.29$\pm$0.50} & 85.58$\pm$0.82 & 85.66$\pm$0.55 \\
		rat vs. sea & 66.59$\pm$2.30 & 74.66$\pm$0.98 & 66.22$\pm$1.77 & 
		75.65$\pm$1.07 & 76.26$\pm$1.04 & \textbf{76.58$\pm$0.99} \\
		\midrule
		平均准确率 & 69.58 & 74.45 & 72.97 & 82.26 & 83.98 & \textbf{84.46} \\
		平均耗时 & 3.03  & 63.64 & 0.98 & 5.32 & \textbf{0.20} & 2.07 \\
		平均排名 & 5.18  & 4.18  & 5.07  & 2.42  & 2.38  & \textbf{1.78}
	\end{longtable}
\end{flushleft}}

\subsection{定义、定理、引理与证明环境的参考样式}
\newtheorem{lemma}{引理}
\begin{lemma}[$\boldsymbol{\pi}$的稀
	疏性] \label{lemma: sparsity}
	假设\eqref{eq:ppmvthsvm1}的解为$\boldsymbol{c}_{+}^{A*}, 
	\boldsymbol{c}_{+}^{B*},R_{+}^{{{A}^{2}*}},R_{+}^{{{B}^{2}*}}$，其对偶问题
	\eqref{eq:dpmvthsvm1}的最优解为$\boldsymbol{\pi}^{*} 
	=\left(\boldsymbol{\alpha 
	}_{+}^{A\top*},\boldsymbol{\alpha }_{+}^{B\top*},\boldsymbol{\beta 
	}^{+\top*},\boldsymbol{\beta }^{-\top*} 
	\right)^{\top}$，则$\forall i \in 
	I^{+}$，多视角训练样本$\boldsymbol{x}_{i} = 
	\left\{x_{i}^{A},x_{i}^{B}\right\}$和$\boldsymbol{\pi}^{*}$
	满足如下关系：
	
样本的$A$视角特征$x_{i}^{A}$和$\boldsymbol{\alpha}_{+}^{A*}$之间满足:
	\begin{align}\ 
	&  x_{i}^{A} \in \mathcal{R}_{+}^{A} = \left\{x^{A} \left| {{\left\| 
	{{\varphi 
	}_{A}}( x^{A} )-\boldsymbol{c}_{+}^{A*} 
	\right\|}^{2}} < R_{+}^{{{A}^{2}*}} \right. \right\} \Rightarrow 
	\alpha_{i}^{A*} = 0, \\
	& x_{i}^{A} \in \mathcal{E}_{+}^{A} = \left\{x^{A} \left| {{\left\| 
	{{\varphi 
				}_{A}}( x^{A} )-\boldsymbol{c}_{+}^{A*} 
			\right\|}^{2}} = R_{+}^{{{A}^{2}*}} \right. \right\} \Rightarrow 
	\alpha_{i}^{A*} \in \left[0, \cfrac{c_{1}^{A}}{l^{+}}\right], \\
	& x_{i}^{A} \in \mathcal{L}_{+}^{A} = \left\{x^{A} \left| {{\left\| 
	{{\varphi 
				}_{A}}( x^{A} )-\boldsymbol{c}_{+}^{A*} 
			\right\|}^{2}} > R_{+}^{{{A}^{2}*}} \right. \right\} \Rightarrow 
	\alpha_{i}^{A*} = \cfrac{c_{1}^{A}}{l^{+}}, 
	\end{align}

样本的$B$视角特征$x_{i}^{B}$和$\boldsymbol{\alpha}_{+}^{B*}$之间满足:
\begin{align}\ 
	&  x_{i}^{B} \in \mathcal{R}_{+}^{B} = \left\{x^{B} \left| {{\left\| 
			{{\varphi 
				}_{B}}( x^{B} )-\boldsymbol{c}_{+}^{B*} 
			\right\|}^{2}} < R_{+}^{{{B}^{2}*}} \right. \right\} \Rightarrow 
	\alpha_{i}^{B*} = 0, \\
	& x_{i}^{B} \in \mathcal{E}_{+}^{B} = \left\{x^{B} \left| {{\left\| 
			{{\varphi 
				}_{B}}( x^{B} )-\boldsymbol{c}_{+}^{B*} 
			\right\|}^{2}} = R_{+}^{{{B}^{2}*}} \right. \right\} \Rightarrow 
	\alpha_{i}^{B*} \in \left[0, \cfrac{c_{1}^{B}}{l^{+}}\right], \\
	& x_{i}^{B} \in \mathcal{L}_{+}^{B} = \left\{x^{B} \left| {{\left\| 
			{{\varphi 
				}_{B}}( x^{B} )-\boldsymbol{c}_{+}^{B*} 
			\right\|}^{2}} > R_{+}^{{{B}^{2}*}} \right. \right\} \Rightarrow 
	\alpha_{i}^{B*} = \cfrac{c_{1}^{B}}{l^{+}}, 
\end{align}

多视角样本$\boldsymbol{x}_{i} 
=\left\{x_{i}^{A},x_{i}^{B}\right\}$和$\boldsymbol{\beta}^{+*},\boldsymbol{\beta}^{-*}$
之间满足：
\begin{align}
	\normalsize 
	\boldsymbol{x}_{i} & \in \mathcal{B}^{+} = \left\{\boldsymbol{x} \left| 
	{{\left\| 
	{{\varphi }_{A}}( x^{A})-\boldsymbol{c}_{+}^{A*} 
	\right\|}^{2}}-R_{+}^{{{A}^{2}*}} - {{\left\|{{\varphi }_{B}}( x^{B} 
	)-\boldsymbol{c}_{+}^{B*} \right\|}^{2}}+R_{+}^{{{B}^{2}*}}  > \epsilon  
	\right. 
	\right\} \nonumber \\ 
	& \Rightarrow \beta_{i}^{+*} = D_{1},\beta_{i}^{-*} = 0, \\
	\boldsymbol{x}_{i} & \in \mathcal{E}^{+} = \left\{\boldsymbol{x} \left| 
	{{\left\| 
			{{\varphi }_{A}}( x^{A})-\boldsymbol{c}_{+}^{A*} 
			\right\|}^{2}}-R_{+}^{{{A}^{2}*}} - {{\left\|{{\varphi }_{B}}( 
			x^{B} 
			)-\boldsymbol{c}_{+}^{B*} \right\|}^{2}}+R_{+}^{{{B}^{2}*}}  = 
			\epsilon  
	\right. 
	\right\} \nonumber \\ 
	& \Rightarrow \beta_{i}^{+*} = D_{1},\beta_{i}^{-*} = 0, \\
	\boldsymbol{x}_{i} & \in \mathcal{E} = \left\{\boldsymbol{x} \left| 
	\quad
	\left|
	{{\left\| 
			{{\varphi }_{A}}( x^{A})-\boldsymbol{c}_{+}^{A*} 
			\right\|}^{2}}-R_{+}^{{{A}^{2}*}} - {{\left\|{{\varphi }_{B}}( 
			x^{B} 
			)-\boldsymbol{c}_{+}^{B*} \right\|}^{2}}+R_{+}^{{{B}^{2}*}} \right|  
			< \epsilon  
	\right. 
	\right\} \nonumber \\ 
	& \Rightarrow \beta_{i}^{+*} =0,\beta_{i}^{-*} = 0, \\
	\boldsymbol{x}_{i} & \in \mathcal{E}^{-} = \left\{\boldsymbol{x} \left| 
	{{\left\| {{\varphi }_{B}}( x^{B})-\boldsymbol{c}_{+}^{B*} 
	\right\|}^{2}}-R_{+}^{{{B}^{2}*}} - {{\left\|{{\varphi }_{A}}( 
	x^{A} )-\boldsymbol{c}_{+}^{A*} \right\|}^{2}}+R_{+}^{{{A}^{2}*}} = 
	\epsilon  
	\right. 
	\right\} \nonumber \\ 
	& \Rightarrow \beta_{i}^{+*} = 0,\beta_{i}^{-*} = \left[0, D_{1}\right], \\
	\boldsymbol{x}_{i} & \in \mathcal{B}^{-} = \left\{\boldsymbol{x} \left| 
	{{\left\| {{\varphi }_{B}}( x^{B})-\boldsymbol{c}_{+}^{B*} 
			\right\|}^{2}}-R_{+}^{{{B}^{2}*}} - {{\left\|{{\varphi }_{A}}( 
			x^{A} )-\boldsymbol{c}_{+}^{A*} \right\|}^{2}}+R_{+}^{{{A}^{2}*}} 
			> 
	\epsilon  
	\right. 
	\right\} \nonumber \\ 
	& \Rightarrow \beta_{i}^{+*} = 0,\beta_{i}^{-*} = D_{1}, 
\end{align}
\end{lemma}
%进一步地，在{\heiti 命题}\ref{thm:beta}和{\heiti 引理}\ref{lemma: sparsity}成立的
%条件下，我们有关于$\boldsymbol{\beta}^{+*},\boldsymbol{\beta}^{-*}$的进一步推论：
%\newtheorem{corollary}{推论}
%\begin{corollary}
%	当参数$\epsilon>0$时，令$\boldsymbol{u}^{-*} = 
%	\boldsymbol{\beta}^{+*} - 
%	\boldsymbol{\beta}^{-*}$，多视角样本$\boldsymbol{x}_{i} 
%	=\left\{x_{i}^{A},x_{i}^{B}\right\}$和$\boldsymbol{u}^{-*}$
%	之间满足：

{\heiti 引理}\ref{lemma: 
	sparsity}通过引理的标签实现引用

\newtheorem{definition}{定义}
\begin{definition}[多视角边界误差点(multi-view margin errors, MEs)] 多视角边界
误差点指的是，该样本点在所有视角上，其
特征均在超球外，即$\boldsymbol{x}_{i} = \left\{x_{i}^{A},x_{i}^{B}\right\} \in 
\mathcal{L}_{+}^{A} \cap \mathcal{L}_{+}^{B}$。该类样本点的集合(MEs)可表述为：
	\begin{equation}
		\normalsize
		\textit{MEs} = \left\{\boldsymbol{x}_{i} = 
		\left\{x_{i}^{A},x_{i}^{B}\right\} \left| 
		\|\varphi(x_{i}^{A}) - \boldsymbol{c}_{+}^{A*}\| > R_{+}^{A^2*} \ 
		\& \ 
		\|\varphi(x_{i}^{B}) - \boldsymbol{c}_{+}^{B*}\| > R_{+}^{B^2*} \right. 
		, i \in I^{+}
		\right\}.
	\end{equation}
\end{definition}

\begin{proof}
	\begin{align}
		&\sum\limits_{i\in {{MEs}}}{\left( \alpha_{i}^{A}+\alpha _{i}^{B} 
			\right)} \le \sum\limits_{i\in {{I}^{+}}}{\left( 
			\alpha_{i}^{A}+\alpha _{i}^{B} 
		\right)}=\sum\limits_{i\in {\textit{CVs}}}{\left( \alpha_{i}^{A}+\alpha 
		_{i}^{B} \right)}=2 \nonumber \\
		\Rightarrow & |\textit{MEs}|\cfrac{c_{1}^{A}+c_{1}^{B}}{l^{+}} \le 2 
		\le 
		|\textit{CVs}|\cfrac{c_{1}^{A}+c_{1}^{B}}{l^{+}} \nonumber \\
		\Rightarrow & \cfrac{|\textit{MEs}|}{l^{+}} \le 
		\cfrac{2}{c_{1}^{A}+c_{1}^{B}} \le 
		\cfrac{|\textit{CVs}|}{l^{+}} \Rightarrow \ {|\textit{MEs}|} \le 
		\cfrac{2{l^{+}}}{c_{1}^{A}+c_{1}^{B}} \le 
		{|\textit{CVs}|} .\nonumber
	\end{align}
\end{proof}

\clearpage
% 重新设置章节标题样式为居中
\CTEXsetup[format={\centering\zihao{3}\heiti}]{section}
\pagestyle{fancy}
\fancyhead[C]{\zihao{5}\CJKfamily{zhsong}参考文献}
\fancyhead[CE]{\zihao{5}\CJKfamily{zhsong}参考文献}
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\section*{个人简介}
\fancyhead[C]{\zihao{5}\CJKfamily{zhsong}个人简介}
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请填写你的个人简介

\section*{导师简介}
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\section*{获得成果目录清单}
\fancyhead[C]{\zihao{5}\CJKfamily{zhsong}获得成果目录清单}
\addcontentsline{toc}{section}{获得成果目录清单}
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\section*{致谢}
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\addcontentsline{toc}{section}{致谢}
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\end{document}
